# Mean-Standard Deviation Analysis of Two-Asset Portfolios

A core idea in finance is that investments should be assessed by their risk as well as by their return. So plotting mean return against risk (measured by the volatility of returns, usually the standard deviation) compares investment portfolios. Optimal portfolios have the highest return for a given risk.

We have two assets (or in general two random variables, but let's stay concrete):

A portfolio mixes proportions a and 1-a of these assets (0 <= a <= 1). The portfolio mean return is linear:

But the standard deviation is not linear. It is the square root of the variance:

This has useful and interesting consequances. For example, adding a bit of an "inferior" investment can give a better portfolio.

To graph return against risk enter the parameters and click New Plot. Up to five plots can be displayed at one time. The Clear button will remove all plots. To see the parameters for each plot click the Legend on/off button. The Redraw button will refresh the graph. This is useful when a has been changed. A single value of a may be plotted by setting the upper and lower bounds of a to the same value (or leave the upper value blank). To see the value of a point, move your cursor to the desired location then click and hold. Note: On each plot a = 0 (Asset 1 alone) is marked with a black dot. For a = 1 (Asset 2 alone) a black square is used.